The two‐point functions of relativistic quantum field theory are discussed in the framework of distribution theory. A derivation in this framework of the Källén‐Lehmann representation is given. The singularities in x space are studied for the example of the DP function. It is in general not possible to extract a singularity with support on the light cone. It is shown however that DP is at best a measure in any neighborhood of the light cone. Thus Lehmann's statement that DP is at least as singular as the corresponding free‐field function is confirmed.

An attempt was made to determine all possible schemes for global symmetry as representations of groups. All possible schemes which correspond to connected groups were found, but the conclusions about nonconnected groups are incomplete. Physical interpretation of the schemes is discussed, and a detailed summary of group‐theoretic methods is included.

It has been pointed out by Wigner that the consistency requirement between the Lagrange and Heisenberg equations of motion does not uniquely determine the canonical commutation relations, at least for one‐dimensional systems. It is shown here that this ambiguity does not arise in local field theory whose basic equal‐time commutators commute with the translation operator.

The high‐energy behavior for a certain class of Feynman diagrams is determined. The results obtained indicate that Regge‐type behavior of scattering amplitude may be associated with the existence of M‐particle bound states for arbitrary M.

A formal theory of three‐particle scattering in a plane is developed using integral equation methods. Expressions for the scattering amplitude and cross section of elastic and inelastic collisions are derived. The effects of indistinguishability of the colliding particles are discussed.

The asymptotic form at high frequencies of the reciprocal of a two‐time thermodynamicGreen's function is shown to be a constant. This constant is expressed as the exact expectation value of a second (functional) derivative of the Hamiltonian.

The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated, with a view to possible applications to statistical mechanics. The functional is used to describe the inequivalent representations. This functional is calculated for the free Bose gas in a box of volume V, and the limit is then taken as V → ∞. In this way we construct cyclic representations describing an infinite system of particles with a density distribution ρ(k) in momentum space. For a given ρ(k) the operator algebra generated by the φ(f), π(g) is reducible. For the ground‐state representation (all particles in the zero‐momentum state), the representation is a direct integral of irreducible representations (analogous to BCS theory). For finite temperatures the situation is complicated by the occurrence of representations which are not type I. The physical significance of the reducibility of the representations is discussed.

It is argued that the thermal ensemble for the infinite system is a pure state at zero temperature, although there is some ambiguity as to which operators should belong to the algebra of observables. For finite temperatures, the thermal ensemble seems to be a mixture. The case of an interacting Bose gas is considered briefly.

A representation is given of fermion annihilation operators ak which satisfy the usual anticommutation relations {ak, ak′+} = δkk′. In this representation the ak operate on periodic functionf(ζ) of a real variable ζ. The Hilbert space is the set of all square integrable functionsf(ζ) of period 2π, with the scalar product defined by

A one‐dimensional gas of impenetrable point‐particle bosons is considered. An exact expression for the one‐particle density matrix is derived in terms of the Fredholm determinant and resolvent of a certain simple kernel. It is proved that for large r, the density matrix is bounded by const × r−4/π2, and that this implies the absence of a generalized Bose‐Einstein condensation, contrary to a recent approximate calculation by Girardeau, who first defined such a condensation.

We derive a new expansion method for the thermodynamic properties and correlation functions of imperfect gases. When the interparticle potential is repulsive, we obtain an alternating series of upper and lower bounds for the activity, free energy, pressure, entropy, internal energy, and for the correlation functions. These bounds are valid even if there is a transition. If the potential has an attractive part, we can obtain upper bounds for the activity, free energy, internal energy, and the correlation functions.

The exact asymptotic form of the coefficients of some two‐dimensional Ising‐model series is derived. A preliminary comparison with some three‐dimensional series suggests that the asymptotic nature of their coefficients is not inconsistent with the same analytic behavior.

A study is made of a particular system of coupled oscillators to determine how the amount of energy exchange and the recurrence time depend on the parameters of the system, and the accuracy with which these are predicted by the perturbation theory developed in part I of this series. The system consists of N + 1 particles, connected by springs which have a quadratic force term, the strength of which, λi, can vary between different particles (``defects''). The equations of motion for the perfect chain (λi = λ) were solved on a computer for the cases N = 4, λ = ¼, ½, ¾; and N = 9, λ = ½; and supplemented by the previous calculations of Fermi, Pasta, and Ulam, in which N = 32, λ = ¼; 1. The ``energy'' in the linear modes, are determined from these computations and compared with the perturbation theory in the first paper of this series. As was found by Fermi, Pasta, and Ulam, when only E1 is initially excited, then only the first few modes acquire an appreciable amount of energy, after which nearly all the energy returns to the first mode in a time τλ. The second‐order perturbation theory is found to give an accurate estimate (within 15%) of both τλ and the amount of energy exchange for all the above cases, except N = 32, λ = 1, which requires higher‐order analysis. It is shown that the nonergodic behavior of this system does not result simply from the incommensurability of the uncoupled frequencies {ωk}, but also from the particular form of mode interaction and the initial conditions used in all calculations, both of which affect the coupled frequency spectrum {Ωk}. To alter the mode interaction a preliminary study is made of ``imperfect'' chains (variable λi), which directly couples low and high modes. It is found that either a large, or else systematic, variation of λi is necessary to appreciably alter the nonergodic behavior of this system. A theoretical examination of the dependence of the coupled frequencies {Ωk} on the initial conditions shows that the ergodic behavior will, in general, be strongly dependent upon the initial conditions.

In previous papers of this series, a theory was constructed for the description of the statistical properties of the eigenvalues of a random matrix of high order. This paper deduces from the same theoretical model the statistical behavior to be expected for a finite stretch of observed eigenvalues chosen out of a much longer stretch of unobserved ones. In comparing the model with experimental data, we have always to deal with such a finite stretch of observed levels. Three ``statistics'' are investigated; these are quantities which can be computed directly from the observed data, and for which the expectation values and statistical variances can be calculated from the theory. One statistic gives only a precise way of measuring the average level spacing. One provides a test of the model by measuring the degree of long‐range order in the level series. The third provides an independent test of the model by measuring the degree of short‐range order. In Sec. V these methods are applied to a preliminary analysis of some experimental data on neutron capture levels in heavy nuclei. The results are inconclusive. Discrepancies between theory and observation are large, but the discrepancies might be produced either by incompleteness of the data or by incorrectness of the theoretical model.

This paper is divided into three disconnected parts. (i) An identity is proved which establishes an intimate connection between the statistical behavior of eigenvalues of random unitary matrices over the real field and over the quaternion field. It is proved that the joint distribution function of all the eigenvalues of a random unitary self‐dual quaternion matrix of order N is identical with the joint distribution function of a set of N alternateeigenvalues of a random unitary symmetric matrix of order 2N. A corollary of this result is the following: the distribution of spacings between next‐nearest‐neighbor eigenvalues in a real symmetric matrix of large order is identical with the distribution of nearest‐neighbor spacings in a self‐dual Hermitian quaternion matrix of large order. (ii) A conjecture is made which gives an exact analytic formula for the partition function of a finite gas of N point charges free to move on an infinite straight line under the influence of an external harmonic potential. This conjecture is at the same time a statement about the statistical properties of the eigenvalues of Hermitian matrices whose elements are Gaussian random variables. (iii) A list is made of several other problems which remain unsolved in the statistical theory of eigenvalues of random matrices.

For simple Lie groups, matrix elements of vector operators (i.e. operators which transform according to the adjoint representation of the group) within an irreducible (finite‐dimensional) representation, are studied. By use of the Wigner‐Eckart theorem, they are shown to be linear combinations of the matrix elements of a finite number of operators. The number of linearly independent terms is calculated and shown to be at most equal to the rank l of the group. l vector operators are constructed explicitly, among which a basis can be chosen for this decomposition. Okubo's mass formula arises as a consequence.